COMPUTATIONAL AND EXPERIMENTAL MODELING OF

SLURRY BUBBLE COLUMN REACTORS

Grant No. : DE-FG-98FT40117

US Department of EnergyNational Energy Technology Laboratory

University Coal ResearchProgram Manager: Donald Krastman

ANNUAL REPORT

Paul Lam, Principal InvestigatorAssociate Dean, College of Engineering

The University of AkronAkron, OH 44326-3903

Tel. : 330-972-7230Email : plam@uakron.edu

and

Dimitri Gidaspow, Co-Principal InvestigatorDepartment of Chemical and Environmental Engineering

Illinois Institute of TechnologyChicago, IL 60616

Tel. : 312-567-3045 Fax : 312-567-8874Email : gidaspow@iit.edu

September 2000

2

COMPUTATIONAL AND EXPERIMENTAL MODELING OF SLURRY

BUBBLE COLUMN REACTORS

DE-FG-98FT40117

UCR ANNUAL REPORT

Dr. Paul Lam, PI Dr. Dimitri GidaspowCollege of Engineering Dept. of Chemical andThe University of Akron Environmental EngineeringAkron, OH 44326-3903 Illinois Institute of Technology

Chicago, IL 60616Tel. : 336-972-7230 Tel. : 312-567-3045, Fax : 312-567-8874Email: plam@uakron.edu Email: gidaspow@iit.edu

ABSTRACT

The objective if this study was to develop a predictive experimentally verifiedcomputational fluid dynamics (CFD) model for gas-liquid-solid flow. A threedimensional transient computer code for the coupled Navier-Stokes equations for eachphase was develooped. The principal input into the model is the viscosity of theparticulate phase which was determined from a measurement of the random kineticenergy of the 800 micron glass beads and a Brookfield viscometer.

The computed time averaged particle velocities and concentrations agree withPIV measurements of velocities and concentrations, obtained using a combination ofgamma-ray and X-ray densitometers, in a slurry bubble column, operated in the bubbly-coalesced fluidization regime with continuos flow of water. Both the experiment and thesimulation show a down-flow of particles in the center of the column and up-flow nearthe walls and nearly uniform particle concentartion.

Normal and shear Reynolds stresses were constructed from the computedinstantaneous particle velocities. The PIV measurement and the simulation producedinstantaneous particle velocities. The PIV measurement and the simulation producedsimilar nearly flat horizontal profiles of turbulent kinetic energy of particles.

This phase of the work was presented at the Chemical Reaction Engineering VIII:Computational Fluid Dynamics, August 6-11, 2000 in Quebec City, Canada.To understand turbulence in risers, measurements were done in the IIT riser with 530micron glass beads using a PIV technique. The results together with simulations will bepresented at the annual meeting of AIChE in November 2000.

4

OBJECTIVE

This project is a collaborative effort between two universities (The AkronUniversity and Illinois Institute of Technology) and two industries (UOP and EnergyInternational). The overall objective of this research is to develop predictivehydrodynamic models for gas-liquid-solid catalyst reactors using computational fluiddynamics (CFD) techniques. The work plan involves a combination of computational,experimental and theoretical studies with a feedback mechanism to correct modelsdeficiencies. The tasks involve: 1- Development of a CFD code for slurry bubble columnreactors; 2- Optimization; 3- Comparison to reactor data; 4- Development of a threedimensional transient CFD code; 5- Measurement of particle turbulent properties; 6a-Measurement of thermal conductivity of particles in the IIT two story riser; 6b-Measurement of evaporation rates of liquid nitrogen in the IIT riser.

ACCOMPLISHMENTS TO DATE

Our paper describing the basic approach using kinetic theory to predict theturbulence of catalyst particles in a slurry bubble column reactor, has been published in arefereed journal (Wu and Gidaspow,”Hydrodynamic simulation of methanol synthesis ingas-liquid slurry bubble column reactors” , Chem. Eng. Sci., 55, 2000, pp. 537-587). Thecomputed slurry height, gas hold up and rate of methanol production agreed with theDepartment of Energy La Porte pilot plant reactor data. The computed turbulent kineticenergy agreed with IIT measurements using a methanol catalyst and with similarmeasurements at Ohio State University in a bubble column extrapolated to no particles.

We have invented an alternate technique for computing turbulence in a slurrybubble column. It involves direct numerical simulation of the equations of motion withthe measured particular viscosity as an input. We have computed the flow profiles,particle concentration profiles and Reynolds stresses for an IIT slurry bubble column. Wesee good agreement between the computations and the measurements done earlier at IIT.The computations were done using our previous two dimensional three phase code and anewly developed three dimensional version. This work was reported in the Ph.D. thesisby Diana Matonis completed in May 2000 and presented in the CFD conference inQuebec City in August 2000.

Measurements of thermal conductivity of catalyst particles in the IIT riser werecompleted. They were reported in the last annual report, September 1999. The IIT riserwas redesigned to eliminate asymmetries, similarly to Sandia National Laboratory riser,sponsored by the Multiphase Fluid Dynamics Research Consortium. Our CCD camerasystem was used to measure Reynolds stresses for 530 micron glass beads. We areinterpreting the data in terms of effective viscosities as shown in the attached short paper.

Work on the modification of the CFD code to include Fisher-Tropsch kinetics isproceeding cooperatively with The University of Akron.

5

CFD SIMULATION OF FLOW AND TURBULENCEIN A SLURRY BUBBLE COLUMN

Diana Matonis, Dimitri Gidaspow and Mitra Bahary

Department of Chemical and Environmental EngineeringIllinois Institute of Technology, Chicago , IL 60616

ABSTRACT

The objective of this study was to develop a predictive experimentally verifiedcomputational fluid dynamic ( CFD ) model for gas-liquid-solid flow. A threedimensional transient computer code for the coupled Navier-Stokes equations for eachphase was developed. The principal input into the model is the viscosity of the particulatephase which was determined from a measurement of the random kinetic energy of the800 micron glass beads and a Brookfield viscometer.

The computed time averaged particle velocities and concentrations agree with PIVmeasurements of velocities and concentrations, obtained using a combination of gamma-ray and X-ray densitometers, in a slurry bubble column, operated in the bubbly-coalescedfluidization regime with continuos flow of water. Both the experiment and the simulationshow a down-flow of particles in the center of the column and up-flow near the walls andnearly uniform particle concentration.

Normal and shear Reynolds stresses were constructed from the computedinstantaneous particle velocities. The PIV measurement and the simulation producedsimilar nearly flat horizontal profiles of turbulent kinetic energy of particles.

6

INTRODUCTION

Fluidized beds are widely used industrially because the particles can be introduced intoand out of the reactor as a fluid and because of good heat and mass transfer in the reactor.For conversion of synthesis gas into methanol or hydrocarbon liquid fuels, a slurrybubble column reactor has several advantages over a fixed bed reactor (Bechtel Group,1990; Viking Systems International, 1994). Cooling surface requirement is less than in afixed bed reactor. Catalyst deactivation due to carbon formation can be handled bycatalyst withdrawal and removal, whereas replacement of fixed bed catalyst requires ashutdown. In view of these advantages, slurry bubble column reactors have recently(Parkinson, 1997;) become competitive with fixed bed reactors for converting synthesisgas into liquid fuels. Fan (1989) has reviewed other applications of three-phasefluidization.Slurry reactor design is usually done (Bechtel Group, 1999; Viking SystemsInternational, 1994) using hold-up correlations. In the early nineties Tarmy andCoulaloglu (1992) of EXXON showed that there were no three phase hydrodynamicmodels in the literature and that there was a need for such models, as illustrated by thedevelopment of a three phase hydrodynamic model at EXXON presented at 1996Computational Fluid Dynamics in Reaction Engineering Conference (Heard et al., 1996).Today, Computational Fluid Dynamics (CFD) has emerged as a new paradigm formodeling multiphase flow and fluidization, as seen from recent conferences (NICHE,2000; FLUIDIZATION IX, 1998; CFD in Reaction Engineering, 2000), the formation ofan industry-led, Department of Energy Multiphase Fluid Dynamics Research Consortium(Thompson, 1999), which consists of 6 national laboratories, 6 universities and Americanchemical companies, and papers published throughout the world.The term CFD has come to denote simulation using Navier-Stokes equations.

Three types of CFD models are being used in the literature to model gas-solid multiphaseflow and fluidization:1.Viscosity Input Models, where the principal input is an empirical viscosity. Examplesare the papers of Anderson, Sundaresan and Jackson (1995) for bubbling beds, Tsuo andGidaspow (1990) and Benyahia, Arastoopour and Knowlton (1998) for risers.2.Kinetic Theory Based Models, as described in Gidaspow (1994).The most successful example of this model is the prediction of the core-annular regimeby Sinclair and Jackson (1989) for steady developed flow in a riser. Transient simulationsand comparisons to data were done by Samuelsberg and Hjertager (1996) and Mathiesen,et al (2000) for multisize flow.3.K-Epsilon Model., where the K corresponds to the granular temperature equation andepsilon is a dissipation for which another conservation law is required. Its success hasbeen to model turbulence for steady single-phase flow. It appears as an option in mostcommercial CFD codes. Kashiwa and VanderHeyden (1998) are extending this model tomultiphase flow as a part of the Multiphase Fluid Dynamics Consortium, where adiscussion has begun at the quarterly meetings concerning mechanisms of turbulenceproduction and dissipation.

7

In single-phase flow, the most fundamental approach to turbulence is DNS, DirectNumerical Simulation of the Navier-Stokes equations. It was quite successful inpredicting the logarithmic velocity profile for channel flow (Kim, el al 1985) and otherturbulence profiles, but with present computers and solution methods is restricted torelatively low Reynolds numbers, about 10,000.The viscosity input model for multiphaseflow is a method similar to the DNS in single phase flow. With particular inputviscosities, a system of coupled Navier-Stokes equations is solved producinginstantaneous fluctuating velocities. Averaging of these velocities produces the normaland the shear Reynolds stresses for the various phases. Such a computation was recentlydone for a bubble column by Pan, Dudukovic and Chung (2000) using the Los AlamosCFDLIB code. Their comparison to the Particle Image Velocity (PIV) data of Mudde, etal (1997) was quite good.

This paper presents a similar computation for three phases. The computed time averageparticle gas and solids hold-ups and the particle velocities generally agree with themeasurements in a slurry bubble column. The computed horizontal profile of particleturbulent kinetic energy also agrees with the PIV measurements, similar to those ofMudde, et al (1997).

Recently Pfleger, et al (1999) and Krishna, et al (1999) applied the commercial CFX codeto bubble columns using the k-epsilon model, while Grevskott, et al (1996) successfullycompared their computed steady state velocity profiles to their experiments. Li, et al(1999) computed the bubble shape in the three-phase system by using an advectionequation for the bubble surface. Discrete particle methods have also been used forsimulating gas-solid systems (e.g. Xu and Yu, 1997; Kwaguchi, Tanaka and Tsuji, 1998)but have not been applied to slurry bubble columns.

PART I. EXPERIMENTAL BUBBLY COALESCED FLOW REGIME

A. Experimental Setup. The setup used in the bubbly coalesced regime for volumefraction, velocity and viscosity measurement experiments consisted of four major parts:fluidization equipment, densitometers assembly, a high resolution micro-imaging Imeasuring system or a video-digital camera unit, and a Brookfield viscometer. Aschematic diagram of the fluidized bed and video-digital camera unit for velocitymeasurements is shown in Figure 1. The schematic diagram for source-detector-recorder� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �schematically in Figure 2.

B. Fluidization Equipment. A rectangular bed was constructed from transparentacrylic (Plexiglas) sheets to facilitate visual observation and video recording of the bedoperations such as gas bubbling and coalescence, and the mixing and segregation ofsolids. The bed height was 213.36 cm and cross-section was 30.48 cm by 5.08 cm. Acentrifugal pump was connected to the bottom of the bed by a 1.0-inch (2.54 cm)diameter stainless steel pipe. Gas injection nozzles from an air compressor wereconnected to the sides of the bed. Liquid was stored in and recycled back to a fifty-fivegallon storage tank.

8

The liquid and gas distributors were located at the bottom of the bed. Two perforatedPlexiglas plates with many 0.28 cm diameter holes distributed the liquid. They wereplaced at 35.6 cm and 50.8 cm above the bottom of the bed, with 0.25 cm size glass beadparticles filled inside. The gas distributor consisted of six staggered porous tubes of 15.24cm length and 0.28 cm diameter. The fine pores of porous tubes had mean diameter of� � � � � � � � � � � � ! � " � # � � � � $ % & � ' % ! ! � � " � ! ! � � � ( ! � � " � ' ) � ! " � $ � # ! � � ! � � $ * + � * 'distributor plate.

C. Densitometer Assembly. Two densitometers were used alternatively formeasuring the time-averaged volume fractions of three phases at a designated location bymeans of the X-ray and y-ray adsorption techniques. The assembly consisted ofradioactive sources as well as detecting and recording devices and a positioning table. Aschematic diagram of the source, detector and recording devices assembly is shown inFigure 2.

(1)Radioactive Source. The source is a 200-mCi Cu-244 source having 17.8-yearhalf-life. It emitted X-rays with photon energy between 12 and 23 keV. The source wascontained in ceramic enamel, recessed into a stainless steel support with a tungsten alloypacking, and sealed in welded Monel Capsule. The device had brazed Beryllium window., - . / 0 1 2 . 3 4 5 1 6 7 8 / - 9 1 / 1 . : 3 ; < 2 9 = 8 = 7 2 > ? @ 7 - A . B 1 0 3 C 8 6 D 3 7 8 6 D E 1 4 2 . 3 4 - F G G @ H 1 I 3 6 5a half-life of 30 years was used. The source was sealed in a welded, stainless steelcapsule. The source holder was welded, filled with lead, and provided with a shutter toturn off the source. This is the same unit used previously by Seo andGidaspow (1987).

(2)Detecting and Recording Devices. The intensity of the X-ray beam was measuredby using a NaI crystal scintillation detector (Teledyne, ST-82-I/B). It consisted of a 2-mmJ K L M N O P Q R S M T U L V T W J W X J Y Z W [ L J K R Q \ ] ^ T T J K L M N _ W X ` a a L Y T [ L b U c [ Q d c X ^ X V `U W b e L J c T W J W X O J K W L b J W b e L J ` c f J K W ^ X V ` Z W V T [ V e U W J W M J W U Z ` V b c J K W X g V h M X ` e J V a U W J W M J c X(Teledyne, S-44-I/2). The dimensions of the crystal were as follows: 5.08 cm thick and5.08 cm in diameter. The two detectors could be switched for use with different sources.The photomultiplier of the detector was connected sequentially to a preamplifier, anamplifier and a single-channel analyzer, a rate meter, and a 186 IBM compatible personalcomputer. The rate meter has a selector and a 0-100-mV scale range.

(3)Positioning Table. Both the source holder and detector were affixed to either sideof the bed on a movable frame and could be moved anywhere up-or-down or to-and-froby means of an electric motor.

D. Particle Image Velocity (PIV) System. The digital camera technique used tomeasure particle velocities as shown on Figure 1 comprised of the following units:

1) Image Recording and Displaying Devices. A high resolution color video cameraequipped with electronic shutter speed settings ranging from OFF to 1/10000 sec andsuper fine pitch color monitor were used to record and display solid velocities.

9

2) Data Recording Device. A 486 I 33 MHz IBM compatible personal computerwith a micro-imaging board inside and a micro-imaging software. Image-Pro Plus wereused to record and store raw solid velocities data at any given location inside thefluidized bed.

E. Brookfield Viscometer. Brookfield digital viscometer (model LVDV-II+) withspring a torque of 673.7 dyne-cm was used to measure the effective bed viscosities. Thisviscometer can produce twenty different rotational speeds ranging from 0 to 100revolutions per minute (rpm) at four different modes, namely, LV, RV, HA, andHBDVII+.

Experimental Procedure and Interpretation.

A. Fluidization Experiments. The liquid from the storage tank was fed to the bedfrom the bottom of the bed using the centrifugal pump. The gas was fed to the bedthrough a compressor. Both gas and liquid from the top of the bed were directed throughthree openings of 1.0-inch (2.54 cm) diameter back to the storage tank, where the gas wasseparated from the liquid.

In order to achieve a uniform fluidization, the liquid distributor section was designedin such a way that the pressure drop through the distributor section was 10 - 20 % of thetotal bed pressure drop. The gas was distributed in the fluidized bed through the sixstaggered porous tubes.

Air and water were used as the gas and liquid, respectively, in this experiment.Ballotini (leaded glass beads) with an average diameter of 0.8 cm and a density of 2.94g/cm3 were used as the solids. The experimental operating conditions are shown in theTable 1 (Bahary, 1994).

i j k l m n o p q r s t u v l w x y p u p r o v w s u v l w j z { r s | s w } { r s | } p w x v u l o p u p r x ~ s p p p w n x p }to measure porosities of fluidized beds (Miller and Gidaspow, 1992: Seo and Gidaspow,1987; Gidaspow, et al, 1995) and solids concentrations in nonaqueous suspensions(Jayaswal, et al, 1990). These techniques are based on the fact that the liquid, gas and same concept was adopted to measure concentration profiles inside our three phasefluidization systems.

  ¡ ¢ £ ¤ ¥ ¦ ¡ ¢ ¡ ¦ ¡ ¢ § ¢ ¤ § ¡ ¨ ¤ ¢ ¢ © ¢ ¡   ª §  the volume fractions of liquid, gas and the solid phases. The amount of radiation that isabsorbed by a material can be given by the Beer-Bougert-Lambert Law:

where I is the intensity of transmitted radiation, Io is the intensity of incident« ¬ ® ¬ ¯ ® ° ± ² ® ³ ¯ ´ µ ¬ ¯ ¯ µ ± ¶ ¬ ¯ ® ° ± · ° µ ¸ ¸ ® · ® µ ± ¯ ² ¹ ® ³ ¯ ´ µ µ ± ³ ® ¯ º ° ¸ » ¬ ¯ µ « ® ¬ ¼ ² ¬ ± ½ ® ³ ¯ ´ µ ¹ ¬ ¯ ´length.

oI=I exp (- )lκ ρ (1)

10

The logarithmic form of equation 1 for three-phase (gas-liquid-solid) fluidized bedsis

whereA lκ ρ=

¾ ¿ À Á Â Ã Ä Ã Å Æ Ç È Â Ã Æ ¿ È Ã ¿ Ç Æ È É Ä Ã ¾ À Æ ¿ Ê Ç Ë Ì È Â Ã Í Î Ä ¾ É Ë Ä Î Ä ¾ É À Ã ¿ Ç Æ È Ë Ï Ã È Ã Ä Ç Ð ¾ ¿ ÀgÑ lÒ Ó Ô

s are the volume fractions of gas, liquid, and solid phases, respectively. The relationfor volume fractions is:

The coefficients in equations (2) were calculated using the least square errorÕ Ö × Ø Ù Ú Û Ü Ö Ý Þ ß à Õ Ø Ö × á â Ú ã Þ á Õ Ú ß Ù à Ö á ä Ü Þ Ö à Ö Ù Õ ä ß Ý Õ Ø Ö Ú Ù Õ Ö Ù ä Ú Õ å Þ Ö á æ Ú Ù ç ä ß Ý è é Þ á å á Ù æ é Þ á ådensitometers at known concentrations of gas, liquid and solids in three phase mixtures.However, these coefficients were found to have values with 20% of error for X-ray andê ë ì í î ï ï ì ï í ì ï ð ï ñ ò ó

C. Velocity Measurements. In order to get a good visualization of microscopicmovement of particles, a fiber-optic light was reflected on the field of view in the frontand the back of the bed. The field of view in most experiments was a 2 cm x 2 cm area.As the particles were fluidized inside the bed, the camera with a zoom lens 18-108 mmand close up focus transferred its field of view to the monitor with streak lines. Thesestreak lines represented the space traveled by the particles in a given time intervalspecified on the camera. The images were then captured and digitized by a micro-imaging board and analyzed using Image-Pro Plus software. Radial and axial velocitymeasurements were conducted at different locations inside the bed. The velocity vectorwas calculated as,

cos

sin

x

y

Lv

t

Lv

t

α

α

∆=∆∆=∆

ô õ ö ÷ ö ø ù ú û ü õ ö ý ú û ü þ ÿ � ö ü ÷ þ � ö � ö ý ø ú û ü õ ö þ ÿ � � ö � ÷ � � õ � ÷ ú � � ÿ ü þ � ø ü ú û ü õ ö ú ÿ � ö ÷ û ö � �shutter speed, and vx and vy are the vertical and horizontal velocity components,respectively.

D. Viscosity Measurements using Brookfield Viscometer. The viscometer wasplaced at the top of the fluidized bed, and secured over the centerline of the bed. Acylindrical spindle (#1 LV) of 0.9421 cm diameter, 7.493 cm effective length and overallheight of 11.50 cm was used. The cylindrical spindle was attached to the bottom of theviscometer without the guard and was lowered inside the fluidized bed by an extensionwire until it was completely immersed in the mixture during measurements.

ln g g l l s so

IA A A

Iε ε ε = + +

1.0g l sε ε ε+ + =

(2)

(4)

(6)

(5)

(3)

11

The measurements in this experiment were made under LV mode at different speedsbetween of 2 and 20 rpm. At each rotational speed, between 10 and 30 readings weretaken. The calibration of the viscometer-spindle apparatus was done using a Newtonianliquid, namely, water.

E. Granular Temperature Determination. The granular temperature, which is 3/2of the random particle kinetic energy, is obtained from the frequency distribution of theinstantaneous velocities measured with the PIV system. Figure 4 shows typical� � � � � � � � � � � � � � � � � � � 2� � � � � � � � � ! " # $ ! � � % & � ! � # � � � ! ' ' � ( ) " * � $ ( )

2 2 21

3 X y zθ σ σ σ = + +

Since no distributions were measured into the depth of the bed, the z direction, and sincethe variance in the direction of flow is the largest, in the calculation the assumption wasmade that the z direction variance equals the x direction variance. Hence

2 21 23 X y

θ σ σ = +

Experimental Results for Bubbly Coalesced Regime.

A. Phase Hold-Up. From the calibration curves of the x-ray and gammaraydensitometers, the time average values of volume fraction for liquid, gas and solid phaseswere calculated. Tables 2 and 3 represent such a subset of the volume fraction of gas andsolids at varying heights and two different horizontal positions. The particle and gasconcentrations appear to be nearly constant throughout the region. A computer simulationof this system, using the experimental superficial liquid velocity of 2cm/s and gasvelocity of 3.37cm/s, also shows uniformity in solids concentration distribution. Figure 3shows the agreement of the computer simulation with experiment. Computer simulationswill be discussed in detail in the next section.

B. Instantaneous Velocity Distribution. The measured velocity data were analyzedusing frequency distribution plots. The frequency distribution plots for particles verticaland horizontal velocities are shown in Figures 4(a) and 4(b) for three phase fluidized bed.

C. Granular Temperature. Figures 5 shows a graph of the granular temperature,calculated using particle velocity measurements, as a function of horizontaldistance from centerline of the bed at two different heights. The granulartemperature of the 800 micron beads is about 100(cm/s)2, except near theleft wall, where there is a higher velocity and more dilute flow due to theasymmetry in the system. This compares to about 1000 (cm/s)2 for 500micron beads in air, determined in the IIT CFB and about 10 (cm/s)2 for500 micron beads in water measured at IIT. Clearly the gas flow increasedthe turbulence of the system. For 45 micron methanol catalyst particles,Wu and Gidaspw(2000) computed the granular temperature to be between20 and 10 for volume fractions corresponding to 0.1 nad 0.25, respectively.These computations approximately agree with the measurements of Mostofi

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12

(2000). The lower value of the granular temperature is due to the smallerparticle size.

D. Fluid Bed Viscosity. The viscosity of the glass beads in the mixture wasobtained in two ways: 1. from a direct measurement of the viscosity by aBrookfield viscometer and 2. from the measurement of the random particlevelocity using the equation

( )( ) ( ) ( )

1 12

2 225 4 41 1 148 1 5 5

s ss o s o s s s

o

de g e g d

e g

ρ πθ θµ ε ε ρπ

= + + + + + Gidaspow and Huilin (1996) have shown that these two methods give the same valueof the viscosity for 75 micron FCC particles in a riser. The same result holds here,since the large viscosity of the 800 micron beads exceeds the viscosity of water andair. Figure 6 shows that the viscosities are the same within experimental error.

Part II. SIMULATION

Hydrodynamic ModelA transient, isothermal, three-dimensional model for multiphase flow was developed.

The hydrodynamic model uses the principle of mass conservation and momentumbalance for each phase. This approach is similar to that of Soo(1967) for multiphase flowand of Jackson (1985) for fluidization. The equations are similar to Bowen’s (1976)balance laws for multi-component mixtures. The principle difference is the appearanceof the volume fraction of phase “k” denoted by k. The fluid pressure, P, is in the liquid(continuous) phase.

For gas-solid fluidized beds, Bouillard, et al. (1989) have shown that this set ofequations produces essentially the same numerical answers for fluidization as did theearlier conditionally stable model, which has the fluid pressure in both the gas and thesolids phases. In this model (hydrodynamic model B), the drag and the stress relationswere altered to satisfy Archimedes’ buoyancy principle and Darcy’s Law, as illustratedby Jayaswal (1991). Note in Table 4, no volume fraction is put into the liquid gravityterm, while in the gas/solid momentum balance contains the buoyancy term. This is ageneralization of model B for gas-solid systems as discussed by Gidaspow(1994) insection 2.4. For the solid phase Pk, equation 12, consists of the static normal stress anddynamic stress, called the solids pressure, which arises due to the collision of theparticles.

This model is unconditionally well-posed, ie, the characteristics are real and distinctfor one-dimensional transient flow. It does not require the presence of solid’s pressurefor stability and well-posedness.

The numerical method is an extension of Harlow and Amsden’s(1971) method,which was subsequently used in the K-FIX program (Rivard and Torrey, 1977). Thepresent program was developed from Jayaswal’s two-dimensional MICE program(1991); which originated from the K-FIX program (Rivard and Torrey, 1977). To obtainthe numerical solution, the non-uniform computational mesh is used in finite-differencingthe equations based on the ICE, implicit Eularian method (Rivard, 1977; Jayaswal, 1991)with appropriate initial and boundary conditions. Stewart and Wedroff (1984) havecritically reviewed the ICE algorithm and related staggered mesh conservative schemes.The scalar variables are located at the cell center and the vector variables at the cell

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13

0.687Re , Re

0.44, Re 1000

kD kk

D k

24C = [1+0.15 ] for < 1000

ReC for= ≥

boundaries. The momentum equation is solved using a staggered mesh, while thecontinuity equation is solved using a donor cell method.

Table 4 shows the continuity and the separate phase momentum equations for three-dimensional transient three-phase flow. There are nine nonlinear-coupled partialdifferential equations for nine dependent variables. The variables to be computed are the+ , - . / 0 1 2 3 4 5 6 , 7 8 9

phases-1, the liquid phase pressure P, and the phase horizontal, x-direction, and vertical velocity, y-direction components, uphase and vphase. The gradient ofpressure is in the fluid (continuous) phase only. This leads to an unconditionally well-posed problem, as discussed in detail by Gidaspow(1994) and Lyczkowski, et al. (1978).

A value of 10poises times the particle concentration was used throughout thesimulations and the value was obtained by fitting the experimental viscosity values forgiven superficial liquid and gas velocities (Bahary, 1994). The viscous stress terms forthe phases are of the Newtonian form as follows

k

; 10( )k kk kk k

Tk k k

[ ] = 2 [ ] poisesS

1 1 [ ] = [ v + ] - I( v )S

2 3

τ ε εµ µ=

=

∇ ∇ ⋅∇ v:;: ? @

( )( )21

150 1.75 k f f kf k ffk kff k kf k k

v v

dd

ε ρε ε µβ β

ε ψε ψ

−−= = +

A A

B C Dk

14

(16)

( )

( )

132

1

33 3

(1 ) 1 3

2 3 1

klk k l l k l

k l

kl k lk l

klf k k l l

k l

e d d

v v

d d

ςα ε ρ ε ρε ε

βςε ρ ρ

ε ε

≠

+ + + + = − + − +

F F

where

( ) ( )( ) ( )

( )( ) ( ) ( )

( )

1 1 1

1

1 1 1

1

kk l k l k l k l

k

kk

k k lkl

k k l k k

kk

k k l

for

for

χφ φ α φ φ φ φ φ φφ

φχφ φ φς

α φ φ φ χ φφχ

φ φ φ

− + − − + − + ≤ + − = − + − − +

> + − î

lk l

k

kk

k f

dd d

dα

εχε ε

= ≥

=+

Here the gas phase is treated as a particulate phase, since it consists primarily ofsmall bubbles. The quantities are time-average as follows

1( , , ) ( , , , )t tottov x y z v x y z t d t+< >= ∫

After time-averaging the equations of continuity and of motion, additional terms arisein the equation of motion (Bird et al., 1960). These terms are the components of theturbulent momentum flux and are referred to as the Reynolds stresses. The equations usedare summarized in Table 5.

Coordinate system and numerical considerations

The solution of the preceding conservation equations depends on the definition ofboundary conditions for adequate comparison to experiment. The diameter of the leadedglass beads was 0.889cm with a density of 2.49 g/cm3. The viscosity was an input in allsimulations to match experimentally obtained viscosities. A value of 10 times the localsolid volume fraction was used throughout these simulations. Several different inletconditions and grid size variations were prescribed to test the sensitivity of the final flowfield solution. Figure 7, Case FB2d3d in Table 6, illustrates the two-dimensionalcomputational domain. The third dimension, to represent the experimental 5.08 cm depthof the bed, is added with a grid size of 1.02 cm. It will be shown that the two-dimensional simulation can properly represent the flow hydrodynamics and be lesscomputer time intensive. The remainder of this section will be to study the effects of

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15

varying grid size, inlet air bubble diameter and void fraction as represented in Table 4.The left side is taken to be the inlet from left wall to centerline of the horizontal, 15 cm.

Three Dimensional High Flow Simulation: FB2d3d

Flow Field and Averaged Velocity Profiles

Figures 8a and 8b show the three-dimensional time-averaged, 16 to 42 seconds, gasand solid volume fraction contour plots along with the time-averaged velocity vectors atthe x-y plane of 3 cm from the front wall. Figure 8c shows the solid contour plot withcorresponding velocity vectors at a time of 39 seconds in the y-z plane of 17.5 cm fromthe left wall. The computed flow pattern correctly shows gas up flow in the center regionas visually confirmed in the experiment. A video of the experiment and of thesimulations shows that the solid fluctuates upward and downward in the center region.Time-averaged velocities show the solid moving downward in the center as illustrated inFigure 8b. Figure 9 reveals that the two-dimensional time-averaged vertical velocitiesoverlap the three-dimensional time-averaged velocity pattern and both agree well withexperiment. Figure 10 further illustrates that the solids vertical velocity does not changewhen going into the bed. Figure 11 further demonstrates the time-averaged waterdownward vertical velocities in the center region. This down flow in the center producesa rotation or particle vorticity as seen in Figure 8a. Due to the buoyancy air moves uponly.

Reynolds Stresses

The stresses are calculated from the velocity vectors directly using equationspresented in Table 5. The profiles of Figures 12 and 14 and all cases studied show thatthe Reynolds stress peaks in the center, whereas peaks close to the wallsin agreement with Muddle, et al., (1997) and Pan, et al., (2000) for gas-liquid flow only.The explanation of the appearance and diagonal vortical movement by Muddle, et al.,(1997) causes drastic swings in the vertical velocity close to the wall, where the motion isprimarily upward. In the center, the horizontal velocity attains its highest magnitude,contributing the most to the horizontal stresses in the center, but the least by the wall. Atthe left wall, the vertical Reynolds stress is the highest, thus the granular temperatureexhibits this maximum at the wall as shown in Figure 13. From experiment, Figure 5, thesame characteristic experimental maximum turbulence is observed closer to the left wall;even at the lower inlet water and air superficial velocities. Figure 14 shows the stressesplotted into the depth at z-y plane of 12 cm from the left wall. The horizontal stressexhibits the characteristic maximum in the center region, but the vertical Reynolds stressis much flatter. This can be attributed to no vortex formation in the third direction.

Two-Dimensional Low Velocity Simulations

Flow Field and Averaged Velocity Profiles

Figure 15 represents the time-averaged, 15 to 44 seconds, solids contour plot andvelocity vectors for Case FB5. Figure 16a illustrates the time-averaged solids’ contourplot and velocity vectors for Case FB2. The only difference between these two cases is

16

the inlet air bubble diameter and grid size. From Figures 18 through 20 it will be shownthat grid size did not affect the transient behavior of the bed. Figure 15 and 16a show thedifference in the bed expansion is due to the increase in bubble diameter. Figures 16b-cillustrate the instantaneous vortex movement from 14 seconds to 15 seconds for CaseFB2.

Figure 17 shows the power spectrum of the vertical velocity for Case FB2 at 8cmfrom left wall and at a bed height of 11 cm. Muddle, et al., (1997) had found a similarlow frequency peak and no dominant frequency above 1 Hz and Bahary (1995) hasmeasured similar low frequencies. The time-averaged vertical and horizontal velocitiesare presented in Figure 18 and agree with the experimental averages, Figure 4, of 2.3cm/s for the vertical velocity and –4.32cm/s for the horizontal velocity at a Bed Height of9 cm and Horizontal Position of 7.5 cm from Left Wall. Figures 18, 19, and 20 representthe time-averaged velocities for the remaining cases with an inlet bubble diameter of0.01cm. The agreement of the time-averaged velocities of Figures 18, 19 and 20 isexpected, since the grid size is the only thing that varies between these cases. However, anoticeable difference exists in these Figures and the two proceeding Figures, 21 and 22.Figure 21 represents the time-averaged horizontal and vertical particle velocity profilesfor a uniform inlet with only an inlet bubble diameter increase. These profiles are muchflatter due to the lack of vortical structure production. Figure 22 represents the larger airbubble diameter and not symmetric inlet conditions (Case FB5). This velocity plot is notflat; instead, there exists an increase in the velocity in the half of the bed where more gasis injected.

Reynolds Stresses and Granular Temperature

The normal and shear stresses are calculated directly from the velocity vectors andpresented in Table 5. Figures 23, 24 and 25 show how damping of the stresses occurs byan increase in grid size. From Figure 23 to 24, the y-directional grid cell is halved in sizefrom 2.5/4.5 cm to 1 cm. The shape of the curves remains the same, but the stresses aredoubled. The same pattern can be seen when going from Figure 25 to 23, where the x-directional grid cell is halved in length and the stresses are double in size. The stressesappear to be linearly proportional to the grid size.

The granular temperature is compared to experiment in Figure 26 and shows generalagreement. The damping in the stresses can also be seen in the granular temperature asthe mean fluctuation of the velocity decreases so does the granular temperature. Figure27 represents the granular temperature of the larger inlet bubble diameter (Case FB4) andas the time-averaged velocity profile was flattened, so is the granular temperature. Thetest for developed flow, as in DNS for single-phase, was performed on the cases. Figure28 presents the test for developed flow of case FB3. The following equations are used toobtain the curves

( )

( )

3

1

3

1

pressuredrop minus weight of bed

’ ’ principal ReynoldsStress

i ii

i ii

dPg dx

dy

u v

ε ρ

ε ρ

=

=

− − =

< > =

∑∫

∑

(17)

17

The two expressions are not equal because the u and v velocities, see Figure 19, are of thesame order of magnitude due to the vortex formation. The stresses shown in Figure 28are on the order of magnitude of the solid’s pressure, which is approximately equal to

2s s s sP vε ρ=

This value is small as compared to the fluid pressure.

DiscussionComputations of the granular temperature and frequency allow us to speculate

concerning vortex size from dimensional analysis and approximate solutions of Navier-Stokes equations for standing waves(Tolstoy,1973). Characteristic length equals thepseudosonic velocity divided by the major frequency; where the pseudosonic velocityequals the square root of the granular temperature. The granular temperature rangesbetween 50 and 100 from Figure 26 and major frequency is taken to be 0.3 Hz. Enteringthese ranges into the equation gives us vortex sizes of 20 and 30 cm. Hence, in oursystem we expect to have one to two vortices and do not expect any vortices into thedepth. Figure 8 shows that there is no vortex in the third dimension. Figure 16 b-d betterillustrates the instantaneous single and double vortices generated by the code and alsovisible seen in experiment. This fluctuating particle vorticity and the flow of the liquidcause the particle concentration to be uniform throughout the bed. That is in contrast tothe case of no liquid flow, where there is a vertical density gradient (Wu and Gidaspow,2000). Such a catalyst distribution is reasonable. It is usually modeled using asedimentation model (Viking, 1993). In production of gasoline in a FCC riser, thereexists a sharp radial catalyst gradient. In view of this undesirable distribution of particles,other designs such as the downer are being considered. Hence, the discovery of auniform concentration described in this study is of some practical significance. In thefuture, this model will be explored further. Further, its’ principle weakness is in theuncertainty of the bubble size. Figure 15 and 16 show an order of magnitude difference inbed expansion caused by the order of magnitude increase in bubble diameter. Such aneffect is reasonable, since the bed expands a lot more for fine particles or bubbles.

Conclusions1. A transient, three-dimensional computer code for the solutions of the

coupled Navier-Stokes equations for gas-liquid-solid flow was developed.The principal input is the particulate viscosity, which was measured with aBrookfield viscometer and a PIV technique.

2. The computed time-averaged particle velocities and concentrations agreewith measurements done in the slurry bubble column with continuous flowof liquid in the churn-turbulent regime. The particle velocities weremeasured using the PIV technique. The concentrations were determinedusing a combination of gamma-ray and x-ray densitometers. Both theexperiment and the simulations show a downflow of particles in the centerof the column with upflow near the wall. The situation is unlike the case ofno liquid recirculation (Wu and Gidaspow, 2000) where there exist largeinhomogeneities of particles in the bed.

3. Computed instantaneous particle velocities were used to construct normaland shear Reynolds stresses, similar to the procedure in DNS for single-

(18)

18

phase flow. The computed horizontal distributions of granular temperature,the turbulent kinetic energy of particles, agreed with measurements doneusing a PIV technique.

AcknowledgementsThis study was partially supported by Department of Energy Grant No. DE-PS26-98FT98200.

NomenclatureAbbreviation TermCD drag coefficientdk characteristic particulate phase diametere coefficient of restitutiong gravityG solid compressive stress modulusgo radial distribution function at contactP Continuous phase pressurePk Dispersed(particulate) phase pressureRek Reynolds number for phase kt timeu horizontal velocity, x-directionv vertical velocity, y-directionw depth velocity, z-direction

Greek Letterskm interphase momentum transfer coefficient

between k and mk volume fraction of phase k

granular temperatureviscositydensity

k stress

kφ solids’ volume fraction at maximum packingparticle sphericity

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19

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22

TABLE 1. Operating Conditions for Bubbly Coalesced Regime Experiments

Temperature (°C) 23.5Particle Mean Diameter (cm) 0.8Particle Density (g/cm3) 2.94Initial Bed Height (cm) 22 /24Minimum Fluidization Velocity (cm/s) 0.76

Table 2. Measured Phase Hold-up in Bubble Coalesced Regime for Vgas=3.37cm/s at 4 cm fromthe Horizontal Center.

Bed heights, cm2.5 5 7.5 10 12.5 15 17.5

solid 0.25 0.18 0.15 0.16 0.2 0.25 0.2air 0.56 0.4 0.34 0.36 0.36 0.3 0.32

Table 3. Measured Phase Hold-up in Bubble Coalesced Regime for Vgas=3.37cm/s at -13 cmfrom the Horizontal Center.

Bed heights, cm2.5 5 7.5 10 12.5 15 17.5

solid 0.15 0.21 0.08 0.12 0.1 0.15 0.16air 0.5 0.43 0. 4 0.41 0.35 0.41 0.53

23

( ) ( ) ( ), ,, ,

kk k k k k k k k k m mm f g s

f

m l g s fm m k km k

v v v P I gt

v v

εε ρ ε ρ ρ ε ρε

β τ

=

=

=≠

∂ + ∇ ⋅ = − ∇ + − +∂

∑ − + ∇ ⋅

∑G G G G

GHG

Table 4. Governing EquationsContinuity Equation for Each Phase, k=g,l,s

( ) ( ) 0k k k k kvt ε ρ ε ρ∂ + ∇ ⋅ =

∂I

Continuous Phase (Liquid) Momentum Balance

( ) ( )

,

f f f f f f f f

m g s fm m f f

v v v P I gt

v v

ε ρ ε ρ ρ

β τ

=

=

=

∂ + ∇ ⋅ = − ∇ ⋅ + +∂

∑ − + ∇ ⋅

JKJ JLJ

Dispersed Phase (Gas or Solid) Momentum Balance

Viscous Stress Tensor

f

f fk k

Tf ff

[ ] = 2 [ ]S

1 1( v ) [ ] = [ v + ] - IS

2 3

τ ε µ=

∇∇ ∇ ⋅ vMM M

24

Table 5. Equations of the calculated stresses.

( ) ( ) ( ) ( )21’ ’ , , , , , , ( , , )

N tv v v x y z t v x y z t v x y z < >= − < >∑

( ) { }{ }1’ ’ ’ ’ ( , , , ) ( , , ) ( , , , ) ( , , )N tu v v u u x y z t u x y z v x y z t v x y z< >=< >= − < > − < > ∑

( ) ( ) ( ) ( )21’ ’ , , , , , , ( , , )

N iu u u x y z t u x y z t u x y z < >= − < >∑

( ) { }{ }1’ ’ ’ ’ ( , , , ) ( , , ) ( , , , ) ( , , )N tu w w u u x y z t u x y z w x y z t w x y z< >=< >= − < > − < > ∑

( ) ( ) ( ) ( )21’ ’ , , , , , , ( , , )

N iw w w x y z t w x y z t w x y z < >= − < >∑

( ) { }{ }1’ ’ ’ ’ ( , , ) ( , , ) ( , , ) ( , , )N tv w w v v x y z v x y z w x y z w x y z< >=< >= − < > − < > ∑ with N(t) being the number of vectors in the time-average

Table 6. Simulation Cases under Investigation

Case N O P Q R O P Q VLiquidcm/s

VGascm/s

Dair ,cm

left,wate

r

left,gas ,right,wat

er =right,gas

FB2d3d

15*2 18*5.825 8.074 6.078 0.1 0.6 0.4 0.5

FB1 32x1cm 2*2.25,19*4.5

4.04 3.37 0.01 0.5 0.5 0.5

FB2 32x1 31x1,2,3,4,10x5

4.04 3.37 0.01 0.5 0.5 0.5

FB3 14x2.5 2*2.25,19*4.5

4.04 3.37 0.01 0.5 0.5 0.5

FB4 14*2.5 2*2.25,19*4.5

4.04 3.37 0.1 0.5 0.5 0.5

FB5 15*2.032

18*5.623 4.04 3.37 0.1 0.6 0.4 0.5

25

Figure 1. Experimental Schematic Diagram for Three-PhaseFluidization System.

26

27

0.20

0.30

0.40

0.50

0.60

0.5 3.5 6.5 9.5 12.5 15.5 18.5 21.5 24.5

Vertical Distance, cm

Gas

Vol

ume

Fra

ctio

n ComputedExperimental

0.00

0.10

0.20

0.30

1.5 5.5 9.5 13.5 17.5 21.5 25.5

Vertical Distance, cm

Sol

id V

olum

e F

ract

ion

Computed

Experimental

Figure 3 (a),(b). Comparison of measured and computed phase hold-up inbubbly coalesced regime for VL=2.04cm/s and VG=3.37cm/s at 4 cm fromhorizontal center of bed.

(a)

(b)

28

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

-32 -27 -22 -17 -12 -7 -2.1 2.93 7.9 12.9 17.9 22.8

Velocity, cm/s

Fre

quen

cy, 1

/cm

/s

Average=-4.32 cm/sSt. Deviation=15.39cm/s

0

0.02

0.04

0.06

0.08

0.1

0.12

-24 -21 -18 -14 -11 -8 -4 -1 2 6 9 12 15

Velocity, cm/s

Fre

quen

cy, 1

/(cm

/s)

Average=2.3 cm/s

StandardDeviation=11.5cm/s

Figure 4. Experimental Vertical (a) and Horizontal (b) VelocityDistribution of Solids’ at a Bed Height of 9 cm and HorizontalPosition of 7.5 cm from Left Wall.

(a)

(b)

29

30

Figure 6. Viscosities determined with a Brookfield Viscometer andfrom a measurement of random particle oscillations using PIV.

31

Figure 7. Inlet and initial conditions for simulations.

Y, v

Z,w

X, u

32

Depth, cm

Ver

tical

Dis

tanc

e,cm

2 3 4 5 60

10

20

30

40

50

60

70

80

90

0.38

0.33

0.29

0.25

0.20

0.16

0.12

0.07

Figure 8. Time-averaged volume fractions and velocities (a)[top,left] for gas, (b)[bottom,center]for particles and 3 cm from front wall (c)[right] for particles.

33

-40-30-20-10

010203040

0 4 8 12 16 20 24 28Horizontal Distance from Left Wall, cm

Vel

ocit

y, c

m/s

AirSolidWater

-15

-10

-5

0

5

10

15

20

0 4 8 12 16 20 24 28

Horizontal Distance from Left Wall, cm

Vel

ocit

y, c

m/s

1 cm from front wallExpt. at Y = 13 cmTwo Dimensional

-15

-5

5

15

25

35

0 4 8 12 16 20 24 28

Horizontal Distance from Left Wall, cm

Vel

ocit

y, c

m/s

1cm 2 cm3cm 4cm5cm

Figure 9. A comparison of two and three-dimensional vertical particle velocities to PIVmeasurements.

Figure 10. Time-averaged vertical particle velocities for various depths from front plate.

Figure 11. A comparison of the time-averaged verical velocities for gas, particles and liquid atthe center of the vessel.

34

-10

0

10

20

30

40

2 6 10 14 18 22 26

Horiontal Distance from Left Wall, cm

Str

esse

s (c

m/s

)2

0

2

4

6

8

10

12

14

2 8 14 20 26

Horizontal Distance from Left Wall, cm

Gra

nula

r T

empe

ratu

re

(cm

/s)2

z=1 cmz=2 cmz=3 cmz = 4 cm

-5

0

5

10

15

20

-2 -1 0 1 2

Depth, cm

Str

ess(

cm/s

)2

Figure 12. Typical computed Reynlds’ stresses for particles at a bedheight of 11.3 cm.

Figure 13. Granular Temperature (3/2 particle random kinetic energy) at several beddepths.

Figure 14. Variation of Reynolds stresses with depth at a bed height of 11.3 cmand 12 cm from Left Wall.

35

Figure 15. Particle contour and velocityvector plot for Case FB5.

16a

16b 16c 16d

Figure 16. (a) Time-averaged and instantaneous time, (b) 14s (c) 15s (d) 25s (e) 28svolume fraction contour and velocity vector plots for Case FB2 with thecorresponding colormap bar for the volume fractions values.

16e

36

0

1000

2000

3000

4000

0.2 0.6 1.1 1.6 2.0 2.5

Frequency Hz.

Pow

er S

pect

rum

Den

sity

Figure 17. Power Spectrum of the Solid Axial Velocity Profile for CaseFB2.

-8

-3

2

7

12

0 3 5 8 10 13 15 18 20 23 25 28 30

Horizontal Distance from Left Wall, cm

Sol

id V

eloc

ity,

cm

/s

verticalhorizontalExpt. Horizontal Pt.Expt. Vertical Point

-10

-5

0

5

10

15

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Horizontal Distance from Wall, cm

Sol

id V

eloc

ity,

cm

/s Vertical

Horizontal

Figure 18. Time-averaged vertical and horizontal particle velocities for CaseFB2 at a bed height of 9 cm.

Figure 19. Time-averaged vertical and horizontal particle velocities forCase FB3 at a bed height of 9 cm.

37

-10

-5

0

5

10

0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0 24.0 27.0 30.0

Horizontal Position, cm

Vel

ocit

y, c

m/s

VerticalHorizontal

Figure 20. Time-averaged vertical and horizontal particle velocitiesfor Case FB1 at a bed height of 9 cm.

-6

-4-2

02

46

8

0 3 5 8 10 13 15 18 20 23 25 28 30

Horizontal Distance from Left Wall, cm

Sol

id V

eloc

ity,

cm

/s VerticalHorizontalExpt. Vertical PointExpt. Horizontal Pt.

Figure 21. Time-averaged vertical and horizontal particle velocities forCase FB4 at a bed height of 9 cm.

-40

-20

0

20

40

60

2 6 10 14 18 22 26

Distance from Left Wall, cm

Vel

ocit

y, c

m/s

Solidairwater

Figure 22. Time-averaged vertical and horizontal particle velocities for CaseFB5 at a bed height of 9 cm.

38

-20

-15

-10

-5

0

5

10

15

20

25

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Horizontal Distance from Left Wall, cm

Str

esse

s (c

m/s

)2

exptexpt.

-10

0

10

20

30

40

50

60

70

1 4 7 10 13 16 19 22 25 28

Horizontal Distance from Left Wall, cm

Str

esse

s (c

m/s

)2

Figure 23. Comparison of computed normal and Reynolds shear stresses forCase FB1 and experimental points at a bed height of 9 cm.

Figure 24. Comparison of computed normal and Reynolds shear stresses forCase FB2 at a bed height of 9 cm.

0

10

20

30

40

50

3 5 8 10 13 15 18 20 23 25 28 30

Horizontal Distance from Left Wall, cm

Gra

nula

r T

empe

ratu

re

(cm

/s)2

0

100

200

300

400

1 6 11 16 21 26Horizontal Distance From Left Wall, cm

Gra

nula

r T

emp.

(cm

/s)2

Computed at y=9Experimental at y=7

-4

-2

0

2

4

6

8

3 5 8 10 13 15 18 20 23 25 28 30

Horizontal Distance from Left Wall, cm

Str

esse

s (c

m/s

)2

Figure 25. Comparison of computed normal and Reynolds shear stresses forCase FB3 and experimental points at a bed height of 9 cm.

Figure 26. Comparison of computed (Case FB2) and experimentalgranular temperature points at a bed height of 9 cm.

Figure 27. Comparison of computed (Case FB4) granulartemperature at a bed height of 9 cm

40

Figure 28. A test for developed flow for case FB3.

-45

-30

-15

0

15

30

45

-17 -14 -11 -8 -5 -2 1 4 7 10

Horizontal Position, cm

(Pa)

weighted pressuredropweighted shear stress